Detection Theory: Sensory and Decision...

Detection Theory:

Sensory and Decision Processes Lewis O. Harvey, Jr.

Department of Psychology

University of Colorado at Boulder

ResponseSystem

(Observable)

StimulusSystem

(Observable)

The Mind

(Inferred)

InternalRepresentation

SensoryProcesses

DecisionProcesses

Psychology of Perception Lewis O. Harvey, Jr.–Instructor Psychology 4165-100 Clare Sims–Assistant Spring 2009 MUEN D156, 09:30–10:45 TR

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Sensory and Decision Processes

A. Introduction All models of detection and discrimination have at least two psychological

components or processes: the sensory process (which transforms physical stimulation into

internal sensations) and a decision process (which decides on responses based on the output

of the sensory process (Krantz, 1969) as illustrated in Figure 1.

ResponseSystem

(Observable)

StimulusSystem

(Observable)

The Mind

(Inferred)

InternalRepresentation

SensoryProcesses

DecisionProcesses

Figure 1: Detection based on two internal processes: sensory and decision.

One goal of classical psychophysical methods was the determination of a stimulus

threshold. Types of thresholds include detection, discrimination, recognition, and

identification. What is a threshold? The concept of threshold actually has two meanings: One

empirical and one theoretical. Empirically speaking, a threshold is the stimulus level that will

allow the observer to perform a task (detection, discrimination, recognition, or identification)

at some criterion level of performance (75% or 84% correct, for example). Theoretically

speaking, a threshold is property of the detection model’s sensory process.

High Threshold Model: The classical concept of a detection threshold, as

represented in the high threshold model (HTM) of detection, is a stimulus level below which

the stimulus has no effect (as if the stimulus were not there) and above which the stimulus

causes the sensory process to generate an output. The classical psychophysical methods (the

method of limits, the method of adjustment, and the method of constant stimuli) developed


–2–

by Gustav Theodor Fechner (1860) were designed to infer the stimulus value corresponding

to the theoretical threshold from the observed detection performance data. In this theoretical

sense, the stimulus threshold is the stimulus energy that exceeds the theoretical threshold

with a probability of 0.5. Until the 1950s the high threshold model of detection dominated

our conceptualization of the detection process and provided the theoretical basis for the

psychophysical measurement of thresholds.

Signal Detection Theory: In the 1950s, a major theoretical advance was made by

combining detection theory and statistical decision theory. As in the high threshold model,

detection performance is based on a sensory process and a decision process. The sensory

process transforms the physical stimulus energy into an internal representation and the

decision process decides what response to make based on this internal representation. The

response can be a simple yes or no (“yes, the stimulus was present” or “no, the stimulus was

not present”) or a more elaborate response, such as a rating of the confidence that the signal

was present. The two processes are each characterized by at least one parameter: The sensory

process by a sensitivity parameter and the decision process by a response criterion parameter.

It was further realized that estimates of thresholds made by the three classical

psychophysical methods confounded the sensitivity of the sensory process with the response

criterion of the decision process. To measure sensitivity and decision criteria, one needs to

measure two aspects of detection performance. Not only must one measure the conditional

probability that the observer says “yes” when a stimulus is present (the hit rate, or HR) but

also one must measure the conditional probability that the observer says “yes” when a

stimulus is not present (the false alarm rate, or FAR). These conditional probabilities are

shown in Table 1.Within the framework of a detection model, these two performance

measures, HR and FAR, can be used to estimate detection sensitivity and decision criterion.

The specific way in which detection sensitivity and response criterion are computed from the

HR and FAR depends upon the specific model one adopts for the sensory process and for the

decision process. Some of these different models and how to distinguish among them are

discussed in a classic paper by David Krantz (1969). The major two competing models

discussed below are the high threshold model and Gaussian signal detection theory.


–3–

Table 1: Conditional probabilities in the simple detection paradigm.

“Yes” “No”

Signal Present Hit Rate (HR) Miss Rate (MR)

Signal Absent False Alarm Rate (FAR)

Correct Rejection Rate (CRR)

High Threshold Model of Detection The high threshold model (HTM) of detection assumes that the sensory process

contains a sensory threshold. When a stimulus is above threshold, the sensory process

generates an output and as a consequence the decision process says “yes.” On trials when the

stimulus is below threshold (and the sensory process therefore does not generate an output)

the decision process might decide to say “yes” anyway, a guess. In the high threshold model

(HTM) the measures of sensory process sensitivity and decision process guessing rate are

computed from the observed hit rate (HR) and false alarm rate (FAR):

p =HR ! FAR

1 ! FAR Sensitivity of the Sensory Process (1)

g = FAR Guessing Rate of the Decision Process (2)

where p is the probability that the stimulus will exceed the threshold of the sensory process

and g is the guessing rate of the decision process (guessing rate is the decision criterion of the

high threshold model). Equation 1 is also called the correction-for-guessing formula.

The High Threshold Model is not valid: Extensive research testing the validity of

the high threshold model has lead to its rejection: It is not an adequate description of the

detection process and therefore Equations 1 and 2 do not succeed in separating the effects of

sensitivity and response bias (Green & Swets, 1966/1974; Krantz, 1969; Macmillan &

Creelman, 2005; McNicol, 1972; Swets, 1961, 1986a, 1986b, 1996; Swets, Tanner, &

Birdsall, 1961; Wickens, 2002). The reasons for rejecting the high threshold model are

discussed next.

The Receiver Operating Characteristic: One important characteristic of any

detection model is the predicted relationship between the hit rate and the false alarm rate as


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the observer changes decision criteria. The plot of HR as a function of FAR is called an ROC

(receiver operating characteristic). By algebraic rearrangement of Equation 1, the high

threshold model of detection predicts a linear relationship between HR and FAR:

HR = p + 1 ! p( ) "FAR Receiver Operating Characteristic (3)

where p is the sensitivity parameter of the high threshold sensory process. This predicted

ROC is shown in Figure 2. When the hit rate and false alarm rate are measured in a detection

experiment using different degrees of response bias, a bowed-shaped ROC (shown by the

filled circles in Figure 2) is obtained. This bowed-shaped ROC is obviously quite different

from the straight-line relationship predicted by the high threshold model and is one of the

bases for rejecting that model.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Hit R

ate

False Alarm Rate

ROC predicted by thehigh threshold model

Observed Data

Figure 2: The Receiver Operating Characteristic (ROC) predicted by the high threshold model of detection compared with typical data.

C. Signal Detection Theory A widely accepted alternative to the high threshold model was developed in the 1950s

and is called signal detection theory (Harvey, 1992). In this model the sensory process has no

sensory threshold (Swets, 1961; Swets et al., 1961; Tanner & Swets, 1954). The sensory

process is assumed to have a continuous output based on random Gaussian noise and that

when a signal is present the signal combines with that noise. By assumption, the noise


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distribution has a mean, µn, of 0.0 and a standard deviation, !

n, of 1.0. The mean of the

signal-plus-noise distribution, µs, and its standard deviation, !

s, depend upon the sensitivity

of the sensory process and the strength of the signal. These two Gaussian probability

distributions are seen in Figure 3. Models based on other probability distributions are also

possible (Egan, 1975).

0.0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3 4

Pro

babili

ty D

ensity

Output of the Sensory Process (X)

Decision Criterion

noise distributionµ=0.0, !=1.0

signal distributionµ=1.0, !=1.0

Xc

Figure 3: Gaussian probability functions of getting a specific output from the sensory process without and with a signal present. The vertical line is the decision criterion, Xc. Outputs higher than Xc lead to a yes response; those lower or equal to Xc lead to a no response.

Measures of the sensitivity of the sensory process are based on the difference between the

mean output under no signal condition and that under signal condition. When the standard

deviations of the two distributions are equal (!n= !

s= 1) sensitivity may be represented by

! d (pronounced “d-prime”):

! d =µ

s" µ

n( )#

n

Equal-Variance Sensitivity (4)

In the more general case, when !n" !

s, the appropriate measure of sensitivity is d

a (“d-sub-

a”) (Macmillan & Creelman, 2005; Simpson & Fitter, 1973; Swets, 1986a, 1986b) :

da=

µs! µ

n( )

"s

2 + "n

2

2

Unequal-Variance Sensitivity (5)


–6–

Note that in the case when !n=!

s (equal-variance model), d

a= ! d .

The decision process is assumed to adopt one or more decision criteria. The output of the

sensory process on each experimental trial is compared to the decision criterion or criteria to

determine which response to give. In the case of one decision criterion, for example, if the

output of the sensory process equals or exceeds the decision criterion, the observer says “yes,

the signal was present.” If the output of the sensory process is less than this criterion, the

observer says “no, the signal was not present.”

Receiver Operating Characteristic: The ROC predicted by the signal detection

model is shown in the left panel of Figure 4 along with the observed data from Figure 2. The

signal detection prediction is in accord with the observed data. The data shown in Figure 4

are fit by a model having µs= 1, !

s= 1 , with a sensitivity of d

a= 1. The fitting of the model

to the data was done using a maximum-likelihood algorithm: the program, RscorePlus, is

available from the author. The ROC predicted by the signal detection theory model is

anchored at the 0,0 and 1,1 points on the graph. Different values of µs generate a different

ROC. For µs= 0 , the ROC is the positive diagonal extending from (0,0) to (1,1). For µ

s

greater than zero, the ROCs are bowed. As µs increases so does the bowing of the

corresponding ROC as may be seen in the right panel of Figure 4 where the ROCs of four

different values of µs are plotted.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Hit R

ate

False Alarm Rate

ROC predicted by thedual-Gaussian, variable-

criterion signal detection model

Observed Data

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Hit R

ate

False Alarm Rate

3.0

2.0

1.0

0.5

Figure 4: Left panel: Receiver Operating Characteristic (ROC) predicted by Signal Detection Theory compared with typical data. Right Panel: ROCs of four different models. The ROC becomes more bowed as the mean signal strength increases.


–7–

The equation for the signal detection theory ROC is that of a straight line if the HR and FAR

are transformed into z-scores using the quantile function of the unit, normal Gaussian

probability distribution (see Appendix I):

z HR( ) =!n

!s

µs" µ

n( ) +!n

!s

z FAR( ) Signal Detection Theory ROC (6a)

where

z HR( ) and

z FAR( ) are the z-scores of the HR and FAR probabilities computed with

the quantile function (see Appendix I). Equation 6a is linear. Let

a = !n!s( ) µ

s" µ

n( ) , and

let b = !n!s, then:

z HR( ) = a + b ! z FAR( ) Signal Detection Theory ROC (6b)

The values of the y-intercept a and the slope b of this ROC are directly related to the mean

and standard deviation of the signal plus noise distribution:

µs

=a

b+ µ

n Mean of Signal plus Noise (7)

!s=!n

b=1

b Standard Deviation of Signal plus Noise (8)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

Z-S

co

re o

f H

it R

ate

Z-Score of False Alarm Rate

ROC predicted by thedual-Gaussian, variable-

criterion signal detection model

Observed Data

-2.0

-1.0

0.0

1.0

2.0

-2.0 -1.0 0.0 1.0 2.0

Z-S

co

re o

f H

it R

ate

Z-Score of False Alarm Rate

3.0

2.0

1.0

0.5

Figure 5: Left panel: Receiver Operating Characteristic (ROC) predicted by Signal Detection Theory compared with typical data. Right panel: ROCs of four different models. The ROC is farther from the diagonal as the mean signal strength increases.


–8–

Equation 6 predicts that when the hit rate and the false alarm rate are transformed from

probabilities into z-scores, the ROC will be a straight line. The z-score transformation from

probability is made using the Gaussian quantile function (see Appendix I) or from tables that

are in every statistics textbook. Some scientific calculators can compute the transformation.

Short computer subroutines based on published algorithms are also available (Press,

Teukolsky, Vetterling, & Flannery, 2002; Zelen & Severo, 1964). These routines are built

into many spread sheet and graphing programs. The ROC predicted by signal detection

theory is shown in the left panel of Figure 5, along with the observed data from the previous

figures. The actual data are fit quite well by a straight line. The right panel of Figure 5 shows

the four ROCs from Figure 4. As the mean of the signal distribution moves farther from the

noise distribution the z-score ROC moves farther away from the positive diagonal.

Sensitivity of the Sensory Process: Sensitivity may be computed either from the

parameters a and b of the linear ROC equation (after they have been computed from the data)

or from the observed HR and FAR pairs of conditional probability:

da=

2

1 + b2! a (9a)

da=

2

1+ b2! z HR( ) " b ! z FAR( )( ) (General Model) (9b)

In the equal-variance model, Equation 9b reduces to the simple form:

da

= ! d = z HR( ) " z FAR( ) (Equal-Variance Model) (9c)

Criteria of the Decision Process: The decision process decision criterion or criteria may be

expressed in terms of a critical output of the sensory process:

Xc= !z FAR( ) Decision Criterion (10)

The decision process decision criterion may also be expressed in terms of the likelihood ratio

that the signal was present, given a sensory process output of x:


–9–

! =

1

"s2#

$ e% 1

2

x% µs

" s

&

'

( (

)

*

+ +

2

1

"n2#

$ e% 12

x% µ n

"n

&

'

( (

)

*

+ +

2 Likelihood Ratio Decision Criterion (11)

A way of expressing response bias is given in Equation 12 (Macmillan & Creelman, 2005):

c = !z HR( ) + z FAR( )

2 Response Bias (12)

Sensitivity is generally a relatively stable property of the sensory process, but the

decision criterion used by an observer can vary widely from task to task and from time to

time. The decision criterion used is influenced by three factors: The instructions to the

observer; the relative frequency of signal trial and no-signal trails (the a priori probabilities);

and the payoff matrix, the relative cost of making the two types of errors (False Alarms and

Misses) and the relative benefit of making the two types of correct responses (Hits and

Correct Rejections). These three factors can cause the observer to use quite different decision

criteria at different times and if the proper index of sensitivity is not used, changes in

decision criterion will be incorrectly interpreted as changes in sensitivity.

D. More Reasons to Reject the High Threshold Model Figure 6 shows the high threshold sensitivity index p for different values of decision

criteria, for an observer having constant sensitivity. The decision criterion is expressed in

terms of the HTM by g and the SDT by Xc The detection sensitivity p calculated from

Equation 1, is not constant, but changes as a function of decision criterion.


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0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

HTM Guessing Rate (g)

HT

M S

ensitiv

ity (

p)

0.0

0.2

0.4

0.6

0.8

1.0

-4 -3 -2 -1 0 1 2 3 4

HT

M S

ensitiv

ity (

p)

Decision Criterion (Xc)

Figure 6: Sensitivity, p, of HTM sensory process computed from HR and FAR in the "yes-no" paradigm as a function of the guessing rate g (left panel) or the decision criterion, Xc, (right panel). The High Threshold Model predicts that p should remain constant.

Another popular index of sensitivity is overall percent correct (hit rate and correct rejection).

In Figure 7 the percent correct is plotted as a function of decision criterion. One sees in

Figure 7 that percent correct also does not remain constant with changes in decision criterion,

a failure of the theory’s prediction. This failure to remain constant is another reason for

rejecting the high threshold model.

0.4

0.5

0.6

0.7

0.8

-3 -2 -1 0 1 2 3

Perc

ent C

orr

ect

Decision Criterion (Xc) Figure 7: Overall percent correct in a "yes-no" experiment for different decision criteria.

E. Two-Alternative, Forced-Choice Detection Paradigm: In a forced-choice paradigm, two or more stimulus alternatives are presented on each

trial and the subject is forced to pick the alternative that is the target. The alternatives can be

presented successively (temporal forced-choice) or simultaneously in different positions in


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the visual field (spatial forced-choice). Forced-choice methods, especially two-alternative

forced-choice (2AFC), are widely-used as an alternative to the single-interval “yes-no”

paradigm discussed above. Because only one performance index, percent correct, is obtained

from this paradigm, it is not possible to calculate both a detection sensitivity index and a

response criterion index. Detection performance in the 2AFC paradigm is equivalent to an

observer using an unbiased decision criterion, and the percent correct performance can be

predicted from signal detection theory. Percent correct in a 2AFC detection experiment

corresponds to the area under the ROC, Az, obtained when the same stimulus is used in the

yes-no signal detection paradigm. Calculation of da from the 2AFC percent correct is

straightforward:

da = 2 ! z pc( ) (Two-Alternative, Forced-Choice) (12)

where

z pc( ) is the z-score transform of the 2AFC percent correct (Egan, 1975; Green &

Swets, 1966/1974; Macmillan & Creelman, 2005; Simpson & Fitter, 1973). The area under

the ROC for da= 1.0 , illustrated in Figure 2 and the left panel in Figure 4, is 0.76 (the

maximum area of the whole graph is 1.0). By rearranging Equation 13, the area under the

ROC may be computed from da by:

Az

= z!1 d

a

2

"

# $

%

& ' (Two-Alternative, Forced-Choice) (13)

where

z!1( ) is the inverse z-score probability transform that converts a z-score into a

probability using the cumulative distribution function (see Appendix I) of the Gaussian

probability distribution.

F. Summary The classical psychophysical methods of limits, of adjustment, and of constant

stimuli, provide procedures for estimating sensory thresholds. These methods, however, are

not able to properly separate the independent factors of sensitivity and decision criterion.

There is no evidence to support the existence of sensory thresholds, at least in the form these

methods were designed to measure.


–12–

Today there are two methods for measuring an observer’s detection sensitivity

relatively uninfluenced by changes in decision criteria. The first method requires that there be

two types of detection trials: Some containing the signal and some containing no signal. Both

detection sensitivity and response criterion may be calculated from the hit rates and false

alarm rates resulting from the performance in these experiments. The second method is the

forced-choice paradigm, which forces all observers to adopt the same decision criterion.

Either of these methods may be used to measure psychometric functions. The “threshold”

stimulus level corresponds to the stimulus producing a specified level of detection

performance. A da of 1.0 or a 2AFC detection of 0.75 are often used to define threshold, but

other values may be chosen as long as they are made explicit.

One advantage of a detection sensitivity measure which is uncontaminated by

decision criterion is that this measure may be used to predict actual performance in a

detection task under a wide variety of different decision criteria. It is risky and without

justification to assume that the decision criterion an observer adopts in the laboratory is the

same when performing a real-world detection task.

A second advantage is that variability in measured sensitivity is reduced because the

variability due to changes in decision criteria is removed. A comparison of contrast

sensitivity functions measured using the method of adjustment (which is contaminated by

decision criterion) and the two-alternative, forced-choice method (not contaminated by

decision criterion) was reported by Higgins, Jaffe, Coletta, Caruso, and de Monasterio

(1984). The variability of the 2AFC measurements is less than one half those made with the

method of adjustment. This reduction of measurement variability will increase the reliability

of the threshold measures and increase its predictive validity.

The material above concerns the behavior of an ideal observer. There may be

circumstances where less than ideal psychophysical procedures must be employed. Factors

such as testing time, ease of administration, ease of scoring, and cost must be carefully

considered in relationship to the desired reliability, accuracy, and ultimate use to which the

measurements will be put. Finally, it must be recognized that no psychophysical method is

perfect. Observers may make decisions in irrational ways or may try to fake a loss of sensory


–13–

capacity. Care must be taken, regardless of the psychophysical method used to measure

capacity, to detect such malingering. But a properly administered, conceptually rigorous

psychophysical procedure will insure the maximum predictive validity of the measured

sensory capacity.


–14–

Appendix I: Gaussian Probability Distribution

The Gaussian distribution has the these properties (Johnson & Kotz, 1970, Chapter 13):

Domain: -∞ to +∞

Probability Density Function (dnorm() in R):

f x : µ,!( ) =1

! 2"e#0.5 x# µ

!

$

% & &

'

( ) )

2

Cumulative Distribution Function (pnorm() in

R):

F x : µ,!( ) =

1+ erfx " µ

! 2

#

$ % &

'

2

Quantile Function (qnorm() in R):

Q p : µ,!( ) = µ + ! 2 erf"10, 2p "1( )[ ]

Mean: µ

Standard Deviation: !


–15–

References

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Harvey, L. O., Jr. (1992). The critical operating characteristic and the evaluation of expert

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Swets, J. A. (1996). Signal Detection Theory and ROC Analysis in Psychology and

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